Subgroups — C₂×D₁₃×C₄⋊C₁₆ #16015892

Order 3328.

862 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₁₃	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#6090	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11718093	C₂	1	2	yes	maximal, complemented
5	D₁₃×C₄⋊C₁₆	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₁₃	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄²×D₁₃	C₄	1	1	yes	characteristic
10	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
14	C₂²×#6090	C₂²	1	2	yes	—
15	C₂×C₄×#1055	C₄	1	1	yes	characteristic
16	C₂×C₁₆×D₁₃	—	2	8	no	—
17	C₂×C₁₆×D₁₃	C₂²	1	8	yes	—
18	C₄×C₈×D₁₃	C₄	1	4	yes	—
19	C₄×C₈×D₁₃	C₂²	1	4	yes	—
20	C₁₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
21	#11718093	C₂²	1	8	yes	complemented
22	C₄⋊#6090	C₂²	1	4	yes	complemented
23	C₂²×C₄×D₁₃	C₈	1	1	yes	characteristic
24	C₂²×C₄×D₁₃	C₈	1	1	yes	characteristic
25	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₁₃	C₈	1	1	yes	characteristic
30	C₂×C₄×Dic₁₃	C₈	1	1	yes	characteristic
31	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
33	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×D₁₃	—	2	4	no	—
35	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
36	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₁₃	D₄	1	4	yes	—
38	C₂×C₈×D₁₃	Q₈	1	4	yes	—
39	C₂×C₈×D₁₃	C₂³	1	2	yes	—
40	C₄²×D₁₃	C₈	1	4	yes	—
41	C₄²×D₁₃	C₂×C₄	1	4	yes	—
42	C₂×C₁₃×C₁₆	—	2	4	no	—
43	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
44	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₁₃	C₂³	1	2	yes	—
46	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
47	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
48	C₈×Dic₁₃	C₂³	1	2	yes	—
49	C₂×#6090	—	2	4	no	—
50	C₂×#6090	C₂³	1	4	yes	—
51	C₄×#1055	C₂×C₄	1	4	yes	—
52	C₁₆×D₁₃	—	2	16	no	—
53	C₂²×C₄⋊C₁₆	—	13	1	no	maximal, Sylow
54	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
59	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
60	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
61	C₂×C₄×D₁₃	—	2	4	no	—
62	C₂×C₄×D₁₃	C₁₆	1	8	yes	complemented
63	C₂×C₄×D₁₃	C₄⋊C₄	1	4	yes	—
64	C₂×C₄×D₁₃	C₂×C₈	1	4	yes	—
65	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
66	C₂×C₄×D₁₃	C₂×C₈	1	4	yes	—
67	C₂×C₄×D₁₃	M₁₆	1	4	yes	—
68	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
74	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
75	C₄²×C₁₃	C₂×C₈	1	2	yes	—
76	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
77	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
78	C₄×Dic₁₃	C₂×C₈	1	4	yes	—
79	C₄×Dic₁₃	C₂×C₈	1	4	yes	—
80	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
81	C₂×#1055	—	2	2	no	—
82	C₂×#1055	C₄⋊C₄	1	4	yes	—
83	C₂×#1055	C₂²×C₄	1	2	yes	—
84	C₂×#1055	C₂²×C₄	1	2	yes	—
85	C₈×D₁₃	—	2	8	no	—
86	C₈×D₁₃	C₄⋊C₄	1	8	yes	—
87	C₈×D₁₃	C₂×D₄	1	4	yes	—
88	C₈×D₁₃	C₂×Q₈	1	4	yes	—
89	C₁₃×C₁₆	—	2	4	no	—
90	#6090	—	2	4	no	—
91	C₂³×C₁₆	—	13	2	no	—
92	C₂²×C₄×C₈	—	13	1	no	—
93	C₂×C₄⋊C₁₆	—	13	2	no	—
94	C₂×C₄⋊C₁₆	—	13	1	no	—
95	C₂×C₄⋊C₁₆	—	13	8	no	—
96	C₂×C₄⋊C₁₆	D₁₃	1	1	yes	characteristic, direct product factor
97	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
98	C₂²×D₁₃	C₄⋊C₈	1	4	yes	—
99	C₂²×D₁₃	C₂×C₁₆	1	4	yes	—
100	C₂²×D₁₃	M₃₂	1	4	yes	—
101	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	C₂×C₁₆	1	4	yes	complemented
104	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
108	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
109	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×Dic₁₃	—	2	2	no	—
111	C₂×Dic₁₃	C₄⋊C₈	1	4	yes	—
112	C₂×Dic₁₃	C₂×C₁₆	1	4	yes	complemented
113	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
114	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
115	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
116	C₄×D₁₃	—	2	8	no	—
117	C₄×D₁₃	C₄⋊C₈	1	8	yes	—
118	C₄×D₁₃	C₂×C₁₆	1	16	yes	complemented
119	C₄×D₁₃	C₂×C₄⋊C₄	1	4	yes	—
120	C₄×D₁₃	C₂×M₁₆	1	4	yes	—
121	C₈×C₁₃	—	2	2	no	—
122	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
123	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
124	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
125	#1055	—	2	2	no	—
126	#1055	C₂×C₄⋊C₄	1	4	yes	—
127	C₂³×C₈	—	13	1	no	—
128	C₂³×C₈	—	13	1	no	—
129	C₂²×C₄²	—	13	1	no	—
130	C₂²×C₁₆	—	13	2	no	—
131	C₂²×C₁₆	—	13	8	no	—
132	C₂²×C₁₆	—	26	8	no	—
133	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
134	C₂×C₄×C₈	—	13	1	no	—
135	C₂×C₄×C₈	—	13	1	no	—
136	C₂×C₄×C₈	—	13	1	no	—
137	C₂×C₄×C₈	—	13	4	no	—
138	C₂×C₄×C₈	—	13	4	no	—
139	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
140	C₄⋊C₁₆	—	13	8	no	—
141	C₄⋊C₁₆	—	13	4	no	—
142	C₄⋊C₁₆	C₂×D₁₃	1	4	yes	direct product factor
143	C₂²×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
144	C₂²×C₁₃	C₂²×C₁₆	1	2	yes	—
145	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
146	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
147	C₂×D₁₃	C₄⋊C₁₆	1	16	yes	direct product factor
148	C₂×D₁₃	C₂×C₄⋊C₈	1	4	yes	—
149	C₂×D₁₃	C₂²×C₁₆	1	4	yes	—
150	C₂×D₁₃	C₂×M₃₂	1	4	yes	—
151	C₄×C₁₃	—	2	2	no	—
152	C₄×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
153	C₄×C₁₃	C₂²×C₁₆	1	4	yes	complemented
154	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
155	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
156	Dic₁₃	—	2	2	no	—
157	Dic₁₃	C₂×C₄⋊C₈	1	4	yes	—
158	Dic₁₃	C₂²×C₁₆	1	4	yes	complemented
159	C₂³×C₄	—	13	1	no	—
160	C₂³×C₄	—	13	1	no	—
161	C₂³×C₄	—	13	1	no	—
162	C₂²×C₈	—	13	1	no	—
163	C₂²×C₈	—	13	2	no	—
164	C₂²×C₈	—	13	4	no	—
165	C₂²×C₈	—	13	4	no	—
166	C₂²×C₈	—	13	2	no	—
167	C₂²×C₈	—	13	1	no	—
168	C₂²×C₈	—	13	4	no	—
169	C₂²×C₈	—	26	4	no	—
170	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
171	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
172	C₂×C₄²	—	13	1	no	—
173	C₂×C₄²	—	13	1	no	—
174	C₂×C₄²	—	13	1	no	—
175	C₂×C₄²	—	13	4	no	—
176	C₂×C₄²	—	13	4	no	—
177	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
178	C₂×C₁₆	—	2	4	no	—
179	C₂×C₁₆	—	13	4	no	—
180	C₂×C₁₆	—	26	16	no	—
181	C₂×C₁₆	—	26	4	no	—
182	C₂×C₁₆	C₂²×D₁₃	1	4	yes	—
183	C₄×C₈	—	13	4	no	—
184	C₄×C₈	—	13	2	no	—
185	C₄×C₈	—	13	2	no	—
186	C₄×C₈	—	13	4	no	—
187	C₄×C₈	C₄×D₁₃	1	2	yes	—
188	C₄×C₈	C₂²×D₁₃	1	2	yes	—
189	C₂×C₁₃	C₂×C₄⋊C₁₆	1	4	yes	complemented
190	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
191	C₂×C₁₃	C₂³×C₁₆	1	1	yes	characteristic
192	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
193	D₁₃	C₂×C₄⋊C₁₆	1	8	yes	direct product factor
194	C₂⁴	—	13	1	no	—
195	C₂²×C₄	—	13	1	no	—
196	C₂²×C₄	—	13	1	no	—
197	C₂²×C₄	—	13	4	no	—
198	C₂²×C₄	—	13	2	no	—
199	C₂²×C₄	—	13	4	no	—
200	C₂²×C₄	—	13	4	no	—
201	C₂²×C₄	—	13	2	no	—
202	C₂²×C₄	—	13	8	no	—
203	C₂²×C₄	—	13	1	no	—
204	C₂²×C₄	—	13	4	no	—
205	C₂²×C₄	—	26	4	no	—
206	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
207	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	13	4	no	—
211	C₂×C₈	—	13	2	no	—
212	C₂×C₈	—	13	8	no	—
213	C₂×C₈	—	13	4	no	—
214	C₂×C₈	—	13	4	no	—
215	C₂×C₈	—	13	2	no	—
216	C₂×C₈	—	26	8	no	—
217	C₂×C₈	—	26	2	no	—
218	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
219	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
220	C₂×C₈	D₄×D₁₃	1	2	yes	—
221	C₂×C₈	Q₈×D₁₃	1	2	yes	—
222	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
223	C₄²	—	13	4	no	—
224	C₄²	—	13	2	no	—
225	C₄²	—	13	4	no	—
226	C₄²	—	13	2	no	—
227	C₄²	C₈×D₁₃	1	2	yes	—
228	C₄²	C₂×C₄×D₁₃	1	2	yes	—
229	C₁₆	—	2	4	no	—
230	C₁₆	—	26	4	no	—
231	C₁₃	C₂²×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
232	C₂³	—	13	2	no	—
233	C₂³	—	13	4	no	—
234	C₂³	—	13	4	no	—
235	C₂³	—	13	4	no	—
236	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	13	2	no	—
239	C₂×C₄	—	13	2	no	—
240	C₂×C₄	—	13	4	no	—
241	C₂×C₄	—	13	4	no	—
242	C₂×C₄	—	13	4	no	—
243	C₂×C₄	—	13	4	no	—
244	C₂×C₄	—	13	8	no	—
245	C₂×C₄	—	13	16	no	—
246	C₂×C₄	—	13	2	no	—
247	C₂×C₄	—	26	8	no	—
248	C₂×C₄	—	26	2	no	—
249	C₂×C₄	C₁₆×D₁₃	1	4	yes	complemented
250	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
251	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
252	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
253	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
254	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
255	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
256	C₈	—	2	2	no	—
257	C₈	—	13	4	no	—
258	C₈	—	26	2	no	—
259	C₈	C₄⋊C₄×D₁₃	1	2	yes	—
260	C₈	C₂×D₄×D₁₃	1	1	yes	characteristic
261	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
262	C₂²	—	13	4	no	—
263	C₂²	—	13	16	no	—
264	C₂²	—	13	4	no	—
265	C₂²	—	13	4	no	—
266	C₂²	D₁₃×C₄⋊C₈	1	2	yes	—
267	C₂²	C₂×C₁₆×D₁₃	1	2	yes	—
268	C₂²	D₁₃×M₃₂	1	2	yes	—
269	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	13	4	no	—
272	C₄	—	13	4	no	—
273	C₄	—	26	2	no	—
274	C₄	D₁₃×C₄⋊C₈	1	2	yes	—
275	C₄	C₂×C₁₆×D₁₃	1	4	yes	complemented
276	C₄	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
277	C₄	C₂×M₁₆×D₁₃	1	1	yes	characteristic
278	C₂	—	13	8	no	—
279	C₂	D₁₃×C₄⋊C₁₆	1	4	yes	direct product factor
280	C₂	C₂×D₁₃×C₄⋊C₈	1	1	yes	characteristic
281	C₂	C₂²×C₁₆×D₁₃	1	1	yes	characteristic
282	C₂	C₂×D₁₃×M₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₁₃
commutator	C₂×C₁₃	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×C₄⋊C₁₆	C₂
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂³×D₁₃	C₂×C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×C₄⋊C₁₆	D₁₃
hypocenter	C₁₃	C₂²×C₄⋊C₁₆
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₂×C₄⋊#6090	C₂	semidirect
4. C₂×#11718093	C₂	semidirect
5. D₁₃×C₄⋊C₁₆	C₂	direct (N × H)
6. C₂×C₁₃×C₄⋊C₁₆	C₂	semidirect
20. C₁₃×C₄⋊C₁₆	C₂²	semidirect
21. #11718093	C₂²	semidirect
22. C₄⋊#6090	C₂²	semidirect
62. C₂×C₄×D₁₃	C₁₆	semidirect
96. C₂×C₄⋊C₁₆	D₁₃	direct (N × H)
103. C₂×C₄×C₁₃	C₂×C₁₆	semidirect
112. C₂×Dic₁₃	C₂×C₁₆	semidirect
118. C₄×D₁₃	C₂×C₁₆	semidirect
142. C₄⋊C₁₆	C₂×D₁₃	direct (N × H)
147. C₂×D₁₃	C₄⋊C₁₆	direct (N × H)
153. C₄×C₁₃	C₂²×C₁₆	semidirect
158. Dic₁₃	C₂²×C₁₆	semidirect
189. C₂×C₁₃	C₂×C₄⋊C₁₆	semidirect
193. D₁₃	C₂×C₄⋊C₁₆	direct (N × H)
231. C₁₃	C₂²×C₄⋊C₁₆	semidirect
249. C₂×C₄	C₁₆×D₁₃	semidirect
275. C₄	C₂×C₁₆×D₁₃	semidirect
279. C₂	D₁₃×C₄⋊C₁₆	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₁₆×D₁₃	2
C₂×C₄×C₈×D₁₃	2
C₂×C₄⋊#6090	2
C₂×#11718093	2
D₁₃×C₄⋊C₁₆	2
C₂×C₁₃×C₄⋊C₁₆	2
C₂²×C₄⋊C₁₆	13

Subgroups — C2×D13×C4⋊C16 #16015892

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₁₃×C₄⋊C₁₆ #16015892